Take Off Performance

Take off performance can be predicted using a simple measure of the
acceleration of the aircraft along the runway based on force
equilibrium.

The forces involved will be,

T – Thrust of propulsion system pushing aircraft along runway.

D – Aerodynamic Drag of vehicle resisting the aircraft motion.

F – Rolling resistance friction due to the contact of wheels or
skids on the ground.

During take-off run the imbalance in these forces will produce an
acceleration along the runway.

(1)

where dV/dt is the acceleration along the runway and m is the
mass of the vehicle.

1. Rotation Velocity.

The
procedure for take-off will be that the vehicle will accelerate until
it reaches a safe initial flying speed, the pilot can then rotate the
vehicle to an attitude to produce climb lift and it will ascend from
the ground. The determination of this safe flying speed or rotation
speed, VR, is a critical factor in determining
take-off performance.

For
safety reasons VR is usually determined as being 1.2 * VSTALL
or 1.1*VMIN CONTROL which ever is greater. Stall
speed, VSTALL, is the lowest speed that the
aircraft can be flown before the airflow starts to separate from
wings as the angle of attack becomes too great. It can be calculated
based on knowledge of the aircraft take-off configuration and hence
the maximum achievable lift coefficient CL(max). To
maintain level flight the lift produced must equal the weight, at the
point just before stall this leads to the following balance,

hence
stall speed can be calculated as,

Minimum
control speed, VMIN CONTROL is a more complex
calculation and requires knowledge of the stall characteristics of
the tailplane and elevator. For conventional aircraft there is only a
small difference between VRcalculations based on
stall speed or minimum control speed.

As
well as rotation speed there are other safety considerations as shown
in the following figure.

V1- Abort decision speed. Below this speed the take-off can
be safely aborted. After this there will not be sufficient runway
length to allow the aircraft to decelerate to a stop.

V2
– Safe climb speed. Below this speed aircraft cannot attain
sufficient climb rate. Aircraft must climb at a minimum gradient to
avoid obstacles at the end of the runway. With engine failure on
multi-engined aircraft, this speed should still be
achievable.

2. Thrust

The
thrust of gas turbine or turbofan engines will be relatively constant
during take-off. A good assumption is to use the manufacturer's
values for maximum static thrust for take-off calculations.

The
thrust of a propeller driven aircraft can be found from the given
shaft horsepower data for the engine and the use of the equations
using propeller efficiency given in the previous section.

(2)

It
is critical to correctly estimate the propeller efficiency for the
particular aircraft velocity along the runway. At V=0 the efficiency
is 0 and at V=VR the
efficiency will be in the range 50% to 80% depending on the type of
propeller system used.

3. Drag

The resistance to motion due to the air viscosity will give a drag of

(3)

where
CD can be considered constant and calculated using
the formula shown in the previous section Section
2.

4. Rolling Resistance

The
friction between aircraft and runway will be proportional to the
normal force exerted by the aircraft on the runway.

(4)

The
normal force will be the difference between Weight of aircraft and
Lift, the friction coefficient will be typically of a magnitude of
0.02 for a standard tarmac runway.

5. Average
acceleration and distance to rotation.

The
rate of change of velocity can be predicted at any point on the
take-off roll by substituting results (2),(3) and (4) into equation
(1). The subsequent velocity can be found by integrating this result
and the distance traveled found by integrating the velocity.

A
typical acceleration will be dominated by the drag component as
thrust, weight and friction are relatively constant during this
period. This leads to the result shown where acceleration is
inversely proportional to velocity2.

Due
to the quadratic nature of acceleration change, an average value,
for the range of velocities from zero to VR can be
found at the point when
.
This average acceleration can be used to simplify calculations and
the take-off run can be assumed to be a constant acceleration for the
time taken to get from 0 to VR.

This means
and
where t is time from start to point where VR
is attained. Rearranging leads to a relatively simple calculation to
predict take-off distance.

6. Take-Off (Balanced) Field Length.

The
required length of runway will be the sum of the distance required to
get to rotation speed and the extra length required to allow for
rapid braking if the pilot decides to abort take-off at the decision
speed V1. This length will typically be
considerably longer than the distance required to achieve flying
speed.

Distance to V1 can be calculated in a manner similar to that
shown for VR. The calculation of braking distance will require knowledge of the
maximum braking friction coefficient that can be generated by the
aircraft. This information should be available from manufacturer's
data. Braking distance calculations should also be done without any
assumption of reverse thrust from engines as during a take-off abort,
engine power may not be available.